Question

[Lê Song Phương]

 1) Chứng minh rằng với mọi số nguyên dương \(n>1\), ta luôn có: \(\log_n\left(n+1\right)>\log_{n+1}\left(n+2\right)\)

 2) Tìm nguyên hàm \(\int\dfrac{x^3-1}{x^4+x}dx\)

Answer 1

[Nguyễn Việt Lâm]

1.

Ta có:

\(\left(n+1\right)^2=n^2+2n+1>n\left(n+2\right)\)

Lấy logarit 2 vế:

\(ln\left(n+1\right)^2>ln\left[n\left(n+2\right)\right]\)

\(\Rightarrow2ln\left(n+1\right)>ln\left(n\right)+ln\left(n+2\right)\ge2\sqrt{ln\left(n\right).ln\left(n+2\right)}\)

\(\Rightarrow ln^2\left(n+1\right)>ln\left(n\right).ln\left(n+2\right)\)

\(\Rightarrow\dfrac{ln\left(n+1\right)}{ln\left(n\right)}>\dfrac{ln\left(n+2\right)}{ln\left(n+1\right)}\)

\(\Rightarrow log_n\left(n+1\right)>log_{n+1}\left(n+2\right)\)

Answer 2

[Nguyễn Việt Lâm]

2.

\(\int\dfrac{x^3-1}{x^4+x}dx=\int\dfrac{2x^3-\left(x^3+1\right)}{x\left(x^3+1\right)}dx=\int\dfrac{2x^2}{x^3+1}dx-\int\dfrac{1}{x}dx\)

\(=\dfrac{2}{3}\int\dfrac{d\left(x^3+1\right)}{x^3+1}-\int\dfrac{dx}{x}\)

\(=\dfrac{2}{3}ln\left|x^3+1\right|-ln\left|x\right|+C\)